Answer:
4x² -11x +5 = 0
Explanation:
Given α and β are roots of 4x² -5x -1 = 0, you want to find the quadratic equation that has (2-α) and (2-β) as its roots.
Factored form
Factoring out the leading coefficient, we can write the factored form of the given quadratic as ...
4(x -α)(x -β) = 0 = 4x² -5x -1
When the roots are changed from α and β to (2-α) and (2-β), the factored form of the new equation is ...
4(x -(2 -α))(x -(2 -β)) = 0
4(x -2 +α)(x -2 +β) = 0
Negating both factors allows us to write this in a familiar form:
4((2 -x) -α)((2 -x) -β) = 0
Comparing to the original factored quadratic, we see that this is the original with x replaced by (2 -x). Then the quadratic we seek is ...
4(2-x)² -5(2 -x) -1 = 0 . . . . . . . . . . . original with x replaced by (2-x)
4(4 -4x +x²) -10 +5x -1 = 0 . . . . simplify a bit
4x² -11x +5 = 0 . . . . . . . . . . simplify
The quadratic whose roots are (2-α) and (2-β) is 4x² -11x +5 = 0.
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Additional comment
Effectively, the roots and the equation are reflected across the line x=1.